Contractor-renormalization approach to frustrated magnets in a magnetic field

We propose to use the contractor-renormalization (CORE) technique in order to derive effective models for quantum magnets in a magnetic field. CORE is a powerful nonperturbative technique that can reduce the complexity of a given microscopic model by focusing on the low-energy part. We provide a detailed analysis of frustrated spin ladders, which have been widely studied in the past: in particular, we discuss how to choose the building block and emphasize the use of their reduced density matrix. With a good choice of basis, CORE is able to reproduce the existence or not of magnetization plateaux in the whole phase diagram contrary to a usual perturbation theory. We also address the issue of plateau formation in two-dimensional bilayers and point out the analogy between nonfrustrated strongly anisotropic models and frustrated SU(2) ones.

Figure 1

The spin ladder with coupling $J_{\perp }$ on vertical rungs, $J_x$ along the legs, and $J_d$ on the diagonal bonds.

Figure 2

(Color online) Size of the magnetization plateau at $m_z=\frac{1}{2}{m}_{\mathit{sat}}$. Results have been obtained after a finite-size scaling analysis of exact diagonalization data obtained on $2\times L$ ladder (up to $L=20$).

Figure 3

(Color online) Finite-size scaling of the size of the magnetization plateau at $m_z=\frac{1}{2}{m}_{\mathit{sat}}$. Data from the numerical exact diagonalization is shown for an example of a finite plateau ($J_x=0.7$ and $J_d=0.5$) and a vanishing plateau ($J_x=0.4$ and $J_d=0.1$).

Figure 4

(Color online) Magnetization $m_z$ along the field (normalized to its saturation value $m_{\mathit{sat}}$) as a function of magnetic field $h$ on a $2\times 16$ ladder with $J_x=0.7$ and $J_d=0.5$; data from the numerical exact diagonalization has been used.

Figure 5

(Color online) Phase diagram using either perturbation theory or range-2 CORE. Reduced density-matrix calculations on a $2\times 12$ ladder indicate that for large $J_x$ and $J_d$, the reduced density-matrix weight of the singlet becomes very small while the $S_z=0$ triplet weight increases substantially (region above the blue dash-dotted line, see text for details). For comparison with the exact results, see Fig. 2.

Figure 6

(Color online) For $J_x=0.4$ and $J_d=0.2$, the norm of $h_r^{\mathit{conn}}$ quickly decreases with increasing range $r$, whereas for $J_x=0.8$ and $J_d=0.7$, the norm of $h_r^{\mathit{conn}}$ has the same order of magnitude for ranges $2⩽r⩽8$.

Figure 7

(Color online) With increasing range, the CORE results converge toward the exact result. Parameters are $2\times L$ with $L=14$; $J_x=0.4$ and $J_d=0.2$. The effective model was obtained up to range 5 and then solved on a 14-site chain.

Figure 8

(Color online) Reduced density-matrix weights for the vertical dimer block obtained on a $2\times 12$ ladder as a function of $J_x$ and $J_d$ along a path in the phase diagram. Calculations have been carried out for a magnetization of $1∕2$ of the saturation value.

Figure 9

(Color online) Comparison of energy vs magnetization for $J_x=J_d=1$ on a $2\times 14$ ladder given by ED and various CORE effective Hamiltonians obtained by choosing two triplets on each vertical rung and keeping up to range-$r$ interactions.

Figure 10

(Color online) Comparison of energy vs magnetization for $J_x=0.8$ and $J_d=0.7$ on a $2\times 14$ ladder given by ED and various CORE effective Hamiltonians obtained by choosing one singlet and two triplets on each vertical rung and keeping up to range-$r$ interactions.

Figure 11

(Color online) Reduced density-matrix weights for the horizontal dimer block obtained on a $2\times 12$ ladder as a function of $J_x$ and $J_d$ along a path in the phase diagram. Calculations have been carried out for a magnetization of $1∕2$ of the saturation value.

Figure 12

(Color online) Energy vs magnetization obtained on a $2\times 12$ ladder with isotropic couplings $J_x=J_d=J_{\perp }=1$. CORE calculations are done with vertical rung blocks, keeping $\mid s⟩$ and $\mid t_{+1}⟩$ and including all effective interactions (corresponds to an infinite range), and horizontal dimer blocks, keeping $\mid s⟩$ and $\mid t_{+1}⟩$ and including range-4 and range-6 effective interactions. Exact results (ED) are shown for comparison.

Figure 13

(Color online) Energy of the plaquette states defined in Eq. (11) as a function of the interaction along the ladder $J_x$ and the diagonal interaction $J_d$ (for $h=0$ and $J_{\perp }=1$).

Figure 14

(Color online) Reduced density-matrix weights of the six chosen plaquette states [see Eq. (11)] as a function of $J_x$ and $J_d$, obtained by ED from the exact ground state of a $2\times 12$ ladder with various $S_z^{\mathit{tot}}$: (a) largest weight of the six plaquette states for all $S_z^{\mathit{tot}}$; (b) minimum over $S_z^{\mathit{tot}}$ of the cumulated weight of the six states. See text for details.

Figure 15

(Color online) Comparison of energy vs magnetization given by ED and CORE Hamiltonians keeping up to range-2 or range-4 effective interactions for a $2\times 14$ ladder with various couplings. [(a)–(c)] For CORE, we keep only the states per plaquette, which have the largest reduced density-matrix weights (see Fig. 14). (d) For CORE, we keep six states per plaquette [see Eq. (11)].

Figure 16

(Color online) Comparison of magnetization curves for the $2\times L$ ladder obtained with ED and range-3 CORE calculations with (a) three kept plaquette states for $J_x=1$ and $J_d=0$, where there is no plateau, and (b) six kept plaquette states for $J_x=0.7$ and $J_d=0.5$, where there is a finite plateau.

Figure 17

(Color online) Critical anisotropy ${\Delta }_c$ above which there is a plateau phase as a function of $J^{\prime }$ for the perturbation theory and the simplest CORE approach for $J=1$. The QMC point, $J^{\prime }∕J=0.29$ and ${\Delta }_c=3.2$, is taken from Ref. 39.

Figure 18

(Color online) For fixed $J=1$ and $J^{\prime }=0.3$: $t$ and $V$ as functions of anisotropy $\Delta$ for range-2 CORE [see Eq. (15)]; dominant parameters: $t_1^{\left(1\right)}$, $t_1^{\left(1\right)}$, $V_2^{\left(1\right)}$, and $V_2^{\left(2\right)}$ also as functions of $\Delta$ for range-4 CORE [see Eq. (17)].

Figure 19

(Color online) Reduced density-matrix weights for the frustrated 2D Heisenberg bilayer obtained on a $2\times 16$ system as a function of $J_x$ and $J_d$ along a path in the phase diagram. Calculations have been carried out for a magnetization of $1∕2$ of the saturation value.

Figure 20

(Color online) For fixed $J_x=0.3$ in the frustrated case: effective $t$ and $V$ parameters as functions of $J_d$ for range-2 CORE [see Eq. (3)]; $t_1^{\left(1\right)}$ and $V_2^{\left(1\right)}$ also as functions of $J_d$ for range-4 CORE [see Eq. (17)].